Proposition 7

Let $G$ be a Hausdorff topological group which has a completion $\hat{G}$. Then the closures in $\hat{G}$ of the neighbourhoods of the identity element in $G$ form a fundamental system of neighbourhoods of the identity element in $\hat{G}$.

Since $\hat{G}$ is regular, every neighbourhood of the identity element in $\hat{G}$ contains the closure $V$ of an open neighbourhood $U$ of $e$ in $\hat{G}$, and $V$ is also the closure of the trace of $U$ on $G$.

Let $G$ be a group which is not necessarily Hausdorff; let $N = \overline{\{e\}}$, and let $G' = G/N$ be the Hausdorff group associated with $G$ (§ 2, no. 6). If $G'$ has a completion $\hat{G}'$, this completion is called the Hausdorff completion of $G$ and is denoted by $\hat{G}$; $\hat{G}'_d$ (resp. $\hat{G}'_s$) is then the Hausdorff completion (Chapter II, § 3, no. 7) of the uniform space $G_d$ (resp. $G_s$).